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Equation 10 · Part 7 · Selection Accounting: The Price Equation Runs the Economy

=

Var⁡(Sel)=σ2∑isi2(zi−zˉ)2,Ne=1HHI\operatorname{Var}(\mathrm{Sel}) = \sigma^2 \sum_i s_i^2 (z_i - \bar z)^2, \qquad N_e = \frac{1}{\mathrm{HHI}}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Every one of those covariance terms is a realized statistic computed on one historical sample path, and molecular evolution learned the hard way not to trust such statistics without a baseline: Kimura’s neutral theory showed that most measured molecular substitution is drift, not selection, and the field has required a null model ever since before crediting any pattern to adaptation. Proposition 4 supplies the economic equivalent. Assume market shares move by pure noise, uncorrelated with productivity — no firm is being selected for or against — and the expected value of the measured selection term is exactly zero. Its variance is not zero, and it has a closed form: Var⁡(Sel)=σ2∑isi2(zi−zˉ)2,Ne=1HHI\operatorname{Var}(\mathrm{Sel}) = \sigma^2 \sum_i s_i^2 (z_i - \bar z)^2, \qquad N_e = \frac{1}{\mathrm{HHI}}.…

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An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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